quad2m (problem 3.2.1, negative)

Percentage Accurate: 52.1% → 86.2%

Time: 10.7s

Alternatives: 9

Speedup: 1.7×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Initial Program: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))

C

double code(double a, double b_2, double c) {
	return (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
}

Python

def code(a, b_2, c):
	return (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]

Alternative 1: 86.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -7.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)} - b\_2\right)}\\ \mathbf{elif}\;b\_2 \leq -1.25 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -7.4e+99)
   (/ (* c -0.5) b_2)
   (if (<= b_2 -7.5e-93)
     (/ (* a c) (* a (- (sqrt (fma c (- a) (* b_2 b_2))) b_2)))
     (if (<= b_2 -1.25e-122)
       (* -0.5 (/ c b_2))
       (if (<= b_2 1.32e+154)
         (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
         (* -2.0 (/ b_2 a)))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -7.4e+99) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= -7.5e-93) {
		tmp = (a * c) / (a * (sqrt(fma(c, -a, (b_2 * b_2))) - b_2));
	} else if (b_2 <= -1.25e-122) {
		tmp = -0.5 * (c / b_2);
	} else if (b_2 <= 1.32e+154) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -7.4e+99)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= -7.5e-93)
		tmp = Float64(Float64(a * c) / Float64(a * Float64(sqrt(fma(c, Float64(-a), Float64(b_2 * b_2))) - b_2)));
	elseif (b_2 <= -1.25e-122)
		tmp = Float64(-0.5 * Float64(c / b_2));
	elseif (b_2 <= 1.32e+154)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -7.4e+99], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, -7.5e-93], N[(N[(a * c), $MachinePrecision] / N[(a * N[(N[Sqrt[N[(c * (-a) + N[(b$95$2 * b$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, -1.25e-122], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[b$95$2, 1.32e+154], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b_2 < -7.4000000000000002e99

   1. Initial program 6.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b\_2} \]

      4. lower-*.f64 96.3

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   5. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -7.4000000000000002e99 < b_2 < -7.50000000000000034e-93

   1. Initial program 51.5%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 51.3

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 51.3%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}}{a} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}\right)}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}\right)} \]

      6. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\color{blue}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}\right)} \]

      8. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\color{blue}{\frac{\sqrt{1}}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}\right)} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\frac{\color{blue}{1}}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}\right)} \]

      10. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) \cdot \left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}} \cdot \frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \frac{1}{\frac{1}{\color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}}\right)} \]

   6. Applied rewrites 42.1%

      \[\leadsto \color{blue}{\frac{b\_2 \cdot b\_2 - \mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}{a \cdot \left(\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)}} \]

   7. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\color{blue}{a \cdot c}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(a\right), b\_2 \cdot b\_2\right)}\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot a}}{a \cdot \left(\left(\mathsf{neg}\left(b\_2\right)\right) + \sqrt{\mathsf{fma}\left(c, \mathsf{neg}\left(a\right), b\_2 \cdot b\_2\right)}\right)} \]

      2. lower-*.f64 80.5

         \[\leadsto \frac{\color{blue}{c \cdot a}}{a \cdot \left(\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)} \]

   9. Applied rewrites 80.5%

      \[\leadsto \frac{\color{blue}{c \cdot a}}{a \cdot \left(\left(-b\_2\right) + \sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)}\right)} \]

   if -7.50000000000000034e-93 < b_2 < -1.25e-122

   1. Initial program 23.3%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 22.0

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 22.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   6. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 81.3

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   8. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

   if -1.25e-122 < b_2 < 1.31999999999999998e154

   1. Initial program 87.9%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.31999999999999998e154 < b_2

   1. Initial program 45.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 45.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 45.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 97.9

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   7. Applied rewrites 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -7.4 \cdot 10^{+99}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq -7.5 \cdot 10^{-93}:\\ \;\;\;\;\frac{a \cdot c}{a \cdot \left(\sqrt{\mathsf{fma}\left(c, -a, b\_2 \cdot b\_2\right)} - b\_2\right)}\\ \mathbf{elif}\;b\_2 \leq -1.25 \cdot 10^{-122}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e+15)
   (/ (* c -0.5) b_2)
   (if (<= b_2 1.32e+154)
     (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a)
     (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e+15) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.32e+154) {
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-1.15d+15)) then
        tmp = (c * (-0.5d0)) / b_2
    else if (b_2 <= 1.32d+154) then
        tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e+15) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 1.32e+154) {
		tmp = (-b_2 - Math.sqrt(((b_2 * b_2) - (a * c)))) / a;
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -1.15e+15:
		tmp = (c * -0.5) / b_2
	elif b_2 <= 1.32e+154:
		tmp = (-b_2 - math.sqrt(((b_2 * b_2) - (a * c)))) / a
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e+15)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 1.32e+154)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(Float64(b_2 * b_2) - Float64(a * c)))) / a);
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -1.15e+15)
		tmp = (c * -0.5) / b_2;
	elseif (b_2 <= 1.32e+154)
		tmp = (-b_2 - sqrt(((b_2 * b_2) - (a * c)))) / a;
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e+15], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 1.32e+154], N[(N[((-b$95$2) - N[Sqrt[N[(N[(b$95$2 * b$95$2), $MachinePrecision] - N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15e15

   1. Initial program 15.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b\_2} \]

      4. lower-*.f64 86.7

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -1.15e15 < b_2 < 1.31999999999999998e154

   1. Initial program 81.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   if 1.31999999999999998e154 < b_2

   1. Initial program 45.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 45.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 45.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 97.9

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   7. Applied rewrites 97.9%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 78.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;\frac{c \cdot -0.5}{b\_2}\\ \mathbf{elif}\;b\_2 \leq 6 \cdot 10^{-153}:\\ \;\;\;\;\frac{\left(-b\_2\right) - \sqrt{c \cdot \left(-a\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -1.15e+15)
   (/ (* c -0.5) b_2)
   (if (<= b_2 6e-153)
     (/ (- (- b_2) (sqrt (* c (- a)))) a)
     (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a))))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -1.15e+15) {
		tmp = (c * -0.5) / b_2;
	} else if (b_2 <= 6e-153) {
		tmp = (-b_2 - sqrt((c * -a))) / a;
	} else {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -1.15e+15)
		tmp = Float64(Float64(c * -0.5) / b_2);
	elseif (b_2 <= 6e-153)
		tmp = Float64(Float64(Float64(-b_2) - sqrt(Float64(c * Float64(-a)))) / a);
	else
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -1.15e+15], N[(N[(c * -0.5), $MachinePrecision] / b$95$2), $MachinePrecision], If[LessEqual[b$95$2, 6e-153], N[(N[((-b$95$2) - N[Sqrt[N[(c * (-a)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b_2 < -1.15e15

   1. Initial program 15.6%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c}{b\_2}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b\_2} \]

      4. lower-*.f64 86.7

         \[\leadsto \frac{\color{blue}{c \cdot -0.5}}{b\_2} \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\frac{c \cdot -0.5}{b\_2}} \]

   if -1.15e15 < b_2 < 6e-153

   1. Initial program 69.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 68.0

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 68.0%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   if 6e-153 < b_2

   1. Initial program 75.0%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 75.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 75.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{c}{b\_2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{c}{b\_2}, \color{blue}{-2 \cdot \frac{b\_2}{a}}\right) \]

      5. lower-/.f64 85.7

         \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \color{blue}{\frac{b\_2}{a}}\right) \]

   7. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* -0.5 (/ c b_2))
   (fma 0.5 (/ c b_2) (* -2.0 (/ b_2 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = fma(0.5, (c / b_2), (-2.0 * (b_2 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = fma(0.5, Float64(c / b_2), Float64(-2.0 * Float64(b_2 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b$95$2), $MachinePrecision] + N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 32.2

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 32.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   6. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 64.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   8. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 76.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   5. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)} \]

      3. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{c}{b\_2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{c}{b\_2}, \color{blue}{-2 \cdot \frac{b\_2}{a}}\right) \]

      5. lower-/.f64 72.8

         \[\leadsto \mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \color{blue}{\frac{b\_2}{a}}\right) \]

   7. Applied rewrites 72.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{c}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \frac{-2}{a}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310)
   (* -0.5 (/ c b_2))
   (fma c (/ 0.5 b_2) (* b_2 (/ -2.0 a)))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = fma(c, (0.5 / b_2), (b_2 * (-2.0 / a)));
	}
	return tmp;
}

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = fma(c, Float64(0.5 / b_2), Float64(b_2 * Float64(-2.0 / a)));
	end
	return tmp
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b$95$2), $MachinePrecision] + N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 32.2

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 32.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   6. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 64.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   8. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in c around 0

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a} + \frac{1}{2} \cdot \frac{c}{b\_2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{c}{b\_2} + -2 \cdot \frac{b\_2}{a}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot c}{b\_2}} + -2 \cdot \frac{b\_2}{a} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{c \cdot \frac{1}{2}}}{b\_2} + -2 \cdot \frac{b\_2}{a} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{c \cdot \frac{\frac{1}{2}}{b\_2}} + -2 \cdot \frac{b\_2}{a} \]

      5. metadata-eval N/A

         \[\leadsto c \cdot \frac{\color{blue}{\frac{1}{2} \cdot 1}}{b\_2} + -2 \cdot \frac{b\_2}{a} \]

      6. associate-*r/ N/A

         \[\leadsto c \cdot \color{blue}{\left(\frac{1}{2} \cdot \frac{1}{b\_2}\right)} + -2 \cdot \frac{b\_2}{a} \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{1}{2} \cdot \frac{1}{b\_2}, -2 \cdot \frac{b\_2}{a}\right)} \]

      8. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2} \cdot 1}{b\_2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(c, \frac{\color{blue}{\frac{1}{2}}}{b\_2}, -2 \cdot \frac{b\_2}{a}\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \color{blue}{\frac{\frac{1}{2}}{b\_2}}, -2 \cdot \frac{b\_2}{a}\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, \color{blue}{\frac{-2 \cdot b\_2}{a}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, \frac{\color{blue}{b\_2 \cdot -2}}{a}\right) \]

      13. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, \color{blue}{b\_2 \cdot \frac{-2}{a}}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \frac{\color{blue}{\mathsf{neg}\left(2\right)}}{a}\right) \]

      15. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{a}\right)\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2 \cdot 1}}{a}\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{a}}\right)\right)\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, \color{blue}{b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)}\right) \]

      19. associate-*r/ N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{a}}\right)\right)\right) \]

      20. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{a}\right)\right)\right) \]

      21. distribute-neg-frac N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \color{blue}{\frac{\mathsf{neg}\left(2\right)}{a}}\right) \]

      22. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(c, \frac{\frac{1}{2}}{b\_2}, b\_2 \cdot \frac{\color{blue}{-2}}{a}\right) \]

      23. lower-/.f64 72.5

          \[\leadsto \mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \color{blue}{\frac{-2}{a}}\right) \]

   5. Applied rewrites 72.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \frac{-2}{a}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(c, \frac{0.5}{b\_2}, b\_2 \cdot \frac{-2}{a}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -0.5 (/ c b_2)) (* -2.0 (/ b_2 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = (-2.0d0) * (b_2 / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = -2.0 * (b_2 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = -2.0 * (b_2 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(-2.0 * Float64(b_2 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = -2.0 * (b_2 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(b$95$2 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 32.2

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 32.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   6. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 64.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   8. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\sqrt{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}{a} \]

      3. flip-- N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}{a} \]

      4. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\frac{1}{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      5. sqrt-div N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{\sqrt{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{\color{blue}{1}}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}{a} \]

      7. lower-/.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \color{blue}{\frac{1}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      8. lower-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\color{blue}{\sqrt{\frac{b\_2 \cdot b\_2 + a \cdot c}{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}}}}}{a} \]

      9. clear-num N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{\frac{\left(b\_2 \cdot b\_2\right) \cdot \left(b\_2 \cdot b\_2\right) - \left(a \cdot c\right) \cdot \left(a \cdot c\right)}{b\_2 \cdot b\_2 + a \cdot c}}}}}}{a} \]

      10. flip-- N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \frac{1}{\sqrt{\frac{1}{\color{blue}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

      12. lower-/.f64 76.0

          \[\leadsto \frac{\left(-b\_2\right) - \frac{1}{\sqrt{\color{blue}{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   4. Applied rewrites 76.0%

      \[\leadsto \frac{\left(-b\_2\right) - \color{blue}{\frac{1}{\sqrt{\frac{1}{b\_2 \cdot b\_2 - a \cdot c}}}}}{a} \]

   5. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]

      2. lower-/.f64 72.5

         \[\leadsto -2 \cdot \color{blue}{\frac{b\_2}{a}} \]

   7. Applied rewrites 72.5%

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{b\_2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (if (<= b_2 -5e-310) (* -0.5 (/ c b_2)) (* b_2 (/ -2.0 a))))

C

double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b_2 <= (-5d-310)) then
        tmp = (-0.5d0) * (c / b_2)
    else
        tmp = b_2 * ((-2.0d0) / a)
    end if
    code = tmp
end function

Java

public static double code(double a, double b_2, double c) {
	double tmp;
	if (b_2 <= -5e-310) {
		tmp = -0.5 * (c / b_2);
	} else {
		tmp = b_2 * (-2.0 / a);
	}
	return tmp;
}

Python

def code(a, b_2, c):
	tmp = 0
	if b_2 <= -5e-310:
		tmp = -0.5 * (c / b_2)
	else:
		tmp = b_2 * (-2.0 / a)
	return tmp

Julia

function code(a, b_2, c)
	tmp = 0.0
	if (b_2 <= -5e-310)
		tmp = Float64(-0.5 * Float64(c / b_2));
	else
		tmp = Float64(b_2 * Float64(-2.0 / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b_2, c)
	tmp = 0.0;
	if (b_2 <= -5e-310)
		tmp = -0.5 * (c / b_2);
	else
		tmp = b_2 * (-2.0 / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b$95$2_, c_] := If[LessEqual[b$95$2, -5e-310], N[(-0.5 * N[(c / b$95$2), $MachinePrecision]), $MachinePrecision], N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b_2 < -4.999999999999985e-310

   1. Initial program 35.8%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around 0

      \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{-1 \cdot \left(a \cdot c\right)}}}{a} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{\left(-1 \cdot a\right) \cdot c}}}{a} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{\color{blue}{c \cdot \left(-1 \cdot a\right)}}}{a} \]

      4. mul-1-neg N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{c \cdot \color{blue}{\left(\mathsf{neg}\left(a\right)\right)}}}{a} \]

      5. lower-neg.f64 32.2

         \[\leadsto \frac{\left(-b\_2\right) - \sqrt{c \cdot \color{blue}{\left(-a\right)}}}{a} \]

   5. Applied rewrites 32.2%

      \[\leadsto \frac{\left(-b\_2\right) - \sqrt{\color{blue}{c \cdot \left(-a\right)}}}{a} \]

   6. Taylor expanded in b_2 around -inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b\_2}} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot \frac{-1}{2}} \]

      3. lower-/.f64 64.9

         \[\leadsto \color{blue}{\frac{c}{b\_2}} \cdot -0.5 \]

   8. Applied rewrites 64.9%

      \[\leadsto \color{blue}{\frac{c}{b\_2} \cdot -0.5} \]

   if -4.999999999999985e-310 < b_2

   1. Initial program 76.1%

      \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
   2. Add Preprocessing

   3. Taylor expanded in b_2 around inf

      \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

      3. associate-/l* N/A

         \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]

      4. metadata-eval N/A

         \[\leadsto b\_2 \cdot \frac{\color{blue}{\mathsf{neg}\left(2\right)}}{a} \]

      5. distribute-neg-frac N/A

         \[\leadsto b\_2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{a}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2 \cdot 1}}{a}\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{a}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \color{blue}{b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)} \]

      9. associate-*r/ N/A

         \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{a}}\right)\right) \]

      10. metadata-eval N/A

          \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{a}\right)\right) \]

      11. distribute-neg-frac N/A

          \[\leadsto b\_2 \cdot \color{blue}{\frac{\mathsf{neg}\left(2\right)}{a}} \]

      12. metadata-eval N/A

          \[\leadsto b\_2 \cdot \frac{\color{blue}{-2}}{a} \]

      13. lower-/.f64 72.3

          \[\leadsto b\_2 \cdot \color{blue}{\frac{-2}{a}} \]

   5. Applied rewrites 72.3%

      \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b\_2 \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b\_2}\\ \mathbf{else}:\\ \;\;\;\;b\_2 \cdot \frac{-2}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ b\_2 \cdot \frac{-2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (* b_2 (/ -2.0 a)))

C

double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = b_2 * ((-2.0d0) / a)
end function

Java

public static double code(double a, double b_2, double c) {
	return b_2 * (-2.0 / a);
}

Python

def code(a, b_2, c):
	return b_2 * (-2.0 / a)

Julia

function code(a, b_2, c)
	return Float64(b_2 * Float64(-2.0 / a))
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = b_2 * (-2.0 / a);
end

Wolfram

code[a_, b$95$2_, c_] := N[(b$95$2 * N[(-2.0 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing

3. Taylor expanded in b_2 around inf

   \[\leadsto \color{blue}{-2 \cdot \frac{b\_2}{a}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{-2 \cdot b\_2}{a}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{b\_2 \cdot -2}}{a} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]

   4. metadata-eval N/A

      \[\leadsto b\_2 \cdot \frac{\color{blue}{\mathsf{neg}\left(2\right)}}{a} \]

   5. distribute-neg-frac N/A

      \[\leadsto b\_2 \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{2}{a}\right)\right)} \]

   6. metadata-eval N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2 \cdot 1}}{a}\right)\right) \]

   7. associate-*r/ N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{2 \cdot \frac{1}{a}}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \color{blue}{b\_2 \cdot \left(\mathsf{neg}\left(2 \cdot \frac{1}{a}\right)\right)} \]

   9. associate-*r/ N/A

      \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\color{blue}{\frac{2 \cdot 1}{a}}\right)\right) \]

   10. metadata-eval N/A

       \[\leadsto b\_2 \cdot \left(\mathsf{neg}\left(\frac{\color{blue}{2}}{a}\right)\right) \]

   11. distribute-neg-frac N/A

       \[\leadsto b\_2 \cdot \color{blue}{\frac{\mathsf{neg}\left(2\right)}{a}} \]

   12. metadata-eval N/A

       \[\leadsto b\_2 \cdot \frac{\color{blue}{-2}}{a} \]

   13. lower-/.f64 38.6

       \[\leadsto b\_2 \cdot \color{blue}{\frac{-2}{a}} \]

5. Applied rewrites 38.6%

   \[\leadsto \color{blue}{b\_2 \cdot \frac{-2}{a}} \]
6. Add Preprocessing

Alternative 9: 2.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{b\_2 \cdot 2}{a} \end{array} \]

FPCore

(FPCore (a b_2 c) :precision binary64 (/ (* b_2 2.0) a))

C

double code(double a, double b_2, double c) {
	return (b_2 * 2.0) / a;
}

Fortran

real(8) function code(a, b_2, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_2
    real(8), intent (in) :: c
    code = (b_2 * 2.0d0) / a
end function

Java

public static double code(double a, double b_2, double c) {
	return (b_2 * 2.0) / a;
}

Python

def code(a, b_2, c):
	return (b_2 * 2.0) / a

Julia

function code(a, b_2, c)
	return Float64(Float64(b_2 * 2.0) / a)
end

MATLAB

function tmp = code(a, b_2, c)
	tmp = (b_2 * 2.0) / a;
end

Wolfram

code[a_, b$95$2_, c_] := N[(N[(b$95$2 * 2.0), $MachinePrecision] / a), $MachinePrecision]

Derivation

1. Initial program 56.6%

   \[\frac{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}{a}} \]

   2. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

   3. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{\left(\mathsf{neg}\left(b\_2\right)\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

   4. lower-/.f64 56.5

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{\left(-b\_2\right) - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

4. Applied rewrites 35.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a}{b\_2 - \sqrt{b\_2 \cdot b\_2 - a \cdot c}}}} \]

5. Taylor expanded in b_2 around -inf

   \[\leadsto \color{blue}{2 \cdot \frac{b\_2}{a}} \]
6. Step-by-step derivation

   1. associate-*r/ N/A

      \[\leadsto \color{blue}{\frac{2 \cdot b\_2}{a}} \]

   2. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 \cdot b\_2}{a}} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{b\_2 \cdot 2}}{a} \]

   4. lower-*.f64 2.6

      \[\leadsto \frac{\color{blue}{b\_2 \cdot 2}}{a} \]

7. Applied rewrites 2.6%

   \[\leadsto \color{blue}{\frac{b\_2 \cdot 2}{a}} \]
8. Add Preprocessing

Developer Target 1: 99.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\left|a\right|} \cdot \sqrt{\left|c\right|}\\ t_1 := \begin{array}{l} \mathbf{if}\;\mathsf{copysign}\left(a, c\right) = a:\\ \;\;\;\;\sqrt{\left|b\_2\right| - t\_0} \cdot \sqrt{\left|b\_2\right| + t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(b\_2, t\_0\right)\\ \end{array}\\ \mathbf{if}\;b\_2 < 0:\\ \;\;\;\;\frac{c}{t\_1 - b\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{b\_2 + t\_1}{-a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b_2 c)
 :precision binary64
 (let* ((t_0 (* (sqrt (fabs a)) (sqrt (fabs c))))
        (t_1
         (if (== (copysign a c) a)
           (* (sqrt (- (fabs b_2) t_0)) (sqrt (+ (fabs b_2) t_0)))
           (hypot b_2 t_0))))
   (if (< b_2 0.0) (/ c (- t_1 b_2)) (/ (+ b_2 t_1) (- a)))))

C

double code(double a, double b_2, double c) {
	double t_0 = sqrt(fabs(a)) * sqrt(fabs(c));
	double tmp;
	if (copysign(a, c) == a) {
		tmp = sqrt((fabs(b_2) - t_0)) * sqrt((fabs(b_2) + t_0));
	} else {
		tmp = hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Java

public static double code(double a, double b_2, double c) {
	double t_0 = Math.sqrt(Math.abs(a)) * Math.sqrt(Math.abs(c));
	double tmp;
	if (Math.copySign(a, c) == a) {
		tmp = Math.sqrt((Math.abs(b_2) - t_0)) * Math.sqrt((Math.abs(b_2) + t_0));
	} else {
		tmp = Math.hypot(b_2, t_0);
	}
	double t_1 = tmp;
	double tmp_1;
	if (b_2 < 0.0) {
		tmp_1 = c / (t_1 - b_2);
	} else {
		tmp_1 = (b_2 + t_1) / -a;
	}
	return tmp_1;
}

Python

def code(a, b_2, c):
	t_0 = math.sqrt(math.fabs(a)) * math.sqrt(math.fabs(c))
	tmp = 0
	if math.copysign(a, c) == a:
		tmp = math.sqrt((math.fabs(b_2) - t_0)) * math.sqrt((math.fabs(b_2) + t_0))
	else:
		tmp = math.hypot(b_2, t_0)
	t_1 = tmp
	tmp_1 = 0
	if b_2 < 0.0:
		tmp_1 = c / (t_1 - b_2)
	else:
		tmp_1 = (b_2 + t_1) / -a
	return tmp_1

Julia

function code(a, b_2, c)
	t_0 = Float64(sqrt(abs(a)) * sqrt(abs(c)))
	tmp = 0.0
	if (copysign(a, c) == a)
		tmp = Float64(sqrt(Float64(abs(b_2) - t_0)) * sqrt(Float64(abs(b_2) + t_0)));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp
	tmp_1 = 0.0
	if (b_2 < 0.0)
		tmp_1 = Float64(c / Float64(t_1 - b_2));
	else
		tmp_1 = Float64(Float64(b_2 + t_1) / Float64(-a));
	end
	return tmp_1
end

MATLAB

function tmp_3 = code(a, b_2, c)
	t_0 = sqrt(abs(a)) * sqrt(abs(c));
	tmp = 0.0;
	if ((sign(c) * abs(a)) == a)
		tmp = sqrt((abs(b_2) - t_0)) * sqrt((abs(b_2) + t_0));
	else
		tmp = hypot(b_2, t_0);
	end
	t_1 = tmp;
	tmp_2 = 0.0;
	if (b_2 < 0.0)
		tmp_2 = c / (t_1 - b_2);
	else
		tmp_2 = (b_2 + t_1) / -a;
	end
	tmp_3 = tmp_2;
end

Wolfram

code[a_, b$95$2_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[Abs[a], $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Abs[c], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = If[Equal[N[With[{TMP1 = Abs[a], TMP2 = Sign[c]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision], a], N[(N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[Abs[b$95$2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Sqrt[b$95$2 ^ 2 + t$95$0 ^ 2], $MachinePrecision]]}, If[Less[b$95$2, 0.0], N[(c / N[(t$95$1 - b$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(b$95$2 + t$95$1), $MachinePrecision] / (-a)), $MachinePrecision]]]]

Reproduce

herbie shell --seed 2024227 
(FPCore (a b_2 c)
  :name "quad2m (problem 3.2.1, negative)"
  :precision binary64
  :herbie-expected 10

  :alt
  (! :herbie-platform default (let ((sqtD (let ((x (* (sqrt (fabs a)) (sqrt (fabs c))))) (if (== (copysign a c) a) (* (sqrt (- (fabs b_2) x)) (sqrt (+ (fabs b_2) x))) (hypot b_2 x))))) (if (< b_2 0) (/ c (- sqtD b_2)) (/ (+ b_2 sqtD) (- a)))))

  (/ (- (- b_2) (sqrt (- (* b_2 b_2) (* a c)))) a))